LMFDB - Embedded newform 336.6.q.j.289.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [336,6,Mod(193,336)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(336, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 0, 4]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("336.193");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 336 = 2^{4} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	336.q (of order $3$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$53.8889634572$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} + 98x^{6} + 83x^{5} + 9122x^{4} - 91x^{3} + 28567x^{2} + 2058x + 86436 $ x^8 - x^7 + 98x^6 + 83x^5 + 9122x^4 - 91x^3 + 28567x^2 + 2058x + 86436
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{8}\cdot 3\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 21)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			289.3
Root			$5.09061 + 8.81720i$ of defining polynomial
Character	$\chi$	$=$	336.289
Dual form			336.6.q.j.193.3

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(4.50000 + 7.79423i) q^{3} +(-11.0358 + 19.1146i) q^{5} +(-126.882 + 26.6059i) q^{7} +(-40.5000 + 70.1481i) q^{9} +O(q^{10})$	\(q+(4.50000 + 7.79423i) q^{3} +(-11.0358 + 19.1146i) q^{5} +(-126.882 + 26.6059i) q^{7} +(-40.5000 + 70.1481i) q^{9} +(208.355 + 360.881i) q^{11} +797.918 q^{13} -198.644 q^{15} +(687.775 + 1191.26i) q^{17} +(1156.51 - 2003.14i) q^{19} +(-778.343 - 869.224i) q^{21} +(-477.701 + 827.402i) q^{23} +(1318.92 + 2284.44i) q^{25} -729.000 q^{27} -7035.29 q^{29} +(630.596 + 1092.22i) q^{31} +(-1875.19 + 3247.93i) q^{33} +(891.689 - 2718.92i) q^{35} +(-4888.22 + 8466.65i) q^{37} +(3590.63 + 6219.16i) q^{39} -5400.95 q^{41} -19686.6 q^{43} +(-893.900 - 1548.28i) q^{45} +(1028.28 - 1781.04i) q^{47} +(15391.3 - 6751.63i) q^{49} +(-6189.98 + 10721.4i) q^{51} +(-9011.37 - 15608.2i) q^{53} -9197.45 q^{55} +20817.2 q^{57} +(3717.84 + 6439.49i) q^{59} +(-1747.69 + 3027.09i) q^{61} +(3272.39 - 9978.09i) q^{63} +(-8805.67 + 15251.9i) q^{65} +(7928.21 + 13732.1i) q^{67} -8598.62 q^{69} -58133.5 q^{71} +(-19555.3 - 33870.8i) q^{73} +(-11870.3 + 20560.0i) q^{75} +(-36038.1 - 40246.0i) q^{77} +(4880.35 - 8453.01i) q^{79} +(-3280.50 - 5681.99i) q^{81} +70395.7 q^{83} -30360.6 q^{85} +(-31658.8 - 54834.7i) q^{87} +(-72153.1 + 124973. i) q^{89} +(-101242. + 21229.3i) q^{91} +(-5675.36 + 9830.01i) q^{93} +(25526.1 + 44212.5i) q^{95} -79328.7 q^{97} -33753.5 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 36 q^{3} - 258 q^{7} - 324 q^{9}+O(q^{10}) $ 8 * q + 36 * q^3 - 258 * q^7 - 324 * q^9	$ 8 q + 36 q^{3} - 258 q^{7} - 324 q^{9} + 402 q^{11} + 924 q^{13} - 276 q^{17} + 510 q^{19} - 3564 q^{21} + 6900 q^{23} - 2814 q^{25} - 5832 q^{27} + 1080 q^{29} - 6410 q^{31} - 3618 q^{33} + 33108 q^{35} - 15250 q^{37} + 4158 q^{39} + 8616 q^{41} - 58396 q^{43} - 15060 q^{47} - 64252 q^{49} + 2484 q^{51} - 13692 q^{53} - 146248 q^{55} + 9180 q^{57} + 34830 q^{59} + 5364 q^{61} - 11178 q^{63} - 66864 q^{65} - 5994 q^{67} + 124200 q^{69} - 178536 q^{71} - 59638 q^{73} + 25326 q^{75} - 75660 q^{77} - 44062 q^{79} - 26244 q^{81} + 416892 q^{83} + 72648 q^{85} + 4860 q^{87} + 77520 q^{89} - 104722 q^{91} + 57690 q^{93} - 221376 q^{95} - 377260 q^{97} - 65124 q^{99}+O(q^{100}) $ 8 * q + 36 * q^3 - 258 * q^7 - 324 * q^9 + 402 * q^11 + 924 * q^13 - 276 * q^17 + 510 * q^19 - 3564 * q^21 + 6900 * q^23 - 2814 * q^25 - 5832 * q^27 + 1080 * q^29 - 6410 * q^31 - 3618 * q^33 + 33108 * q^35 - 15250 * q^37 + 4158 * q^39 + 8616 * q^41 - 58396 * q^43 - 15060 * q^47 - 64252 * q^49 + 2484 * q^51 - 13692 * q^53 - 146248 * q^55 + 9180 * q^57 + 34830 * q^59 + 5364 * q^61 - 11178 * q^63 - 66864 * q^65 - 5994 * q^67 + 124200 * q^69 - 178536 * q^71 - 59638 * q^73 + 25326 * q^75 - 75660 * q^77 - 44062 * q^79 - 26244 * q^81 + 416892 * q^83 + 72648 * q^85 + 4860 * q^87 + 77520 * q^89 - 104722 * q^91 + 57690 * q^93 - 221376 * q^95 - 377260 * q^97 - 65124 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/336\mathbb{Z}\right)^\times$.

$n$	$85$	$113$	$127$	$241$
$\chi(n)$	$1$	$1$	$1$	$e\left(\frac{1}{3}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	4.50000	+	7.79423i	0.288675	+	0.500000i
$3$	4.50000	+	7.79423i	0.288675	+	0.500000i
$4$		0			0
$5$	−11.0358	+	19.1146i	−0.197414	+	0.341932i	−0.947689	−	0.319194i	$-0.896588\pi$
$5$	−11.0358	+	19.1146i	−0.197414	+	0.341932i	0.750275	+	0.661126i	$0.229921\pi$
$6$		0			0
$7$	−126.882	+	26.6059i	−0.978715	+	0.205226i
$7$	−126.882	+	26.6059i	−0.978715	+	0.205226i
$8$		0			0
$9$	−40.5000	+	70.1481i	−0.166667	+	0.288675i
$10$		0			0
$11$	208.355	+	360.881i	0.519184	+	0.899254i	0.999751	+	0.0222959i	$0.00709758\pi$
$11$	208.355	+	360.881i	0.519184	+	0.899254i	−0.480567	+	0.876958i	$0.659569\pi$
$12$		0			0
$13$	797.918			1.30948			0.654742	−	0.755853i	$-0.272777\pi$
$13$	797.918			1.30948			0.654742	+	0.755853i	$0.272777\pi$
$14$		0			0
$15$	−198.644			−0.227955
$16$		0			0
$17$	687.775	+	1191.26i	0.577197	+	0.999735i	0.995799	+	0.0915652i	$0.0291870\pi$
$17$	687.775	+	1191.26i	0.577197	+	0.999735i	−0.418602	+	0.908170i	$0.637480\pi$
$18$		0			0
$19$	1156.51	−	2003.14i	0.734965	−	1.27300i	−0.219774	−	0.975551i	$-0.570532\pi$
$19$	1156.51	−	2003.14i	0.734965	−	1.27300i	0.954739	−	0.297446i	$-0.0961347\pi$
$20$		0			0
$21$	−778.343	−	869.224i	−0.385144	−	0.430114i
$22$		0			0
$23$	−477.701	+	827.402i	−0.188294	+	0.326135i	−0.944682	−	0.327989i	$-0.893629\pi$
$23$	−477.701	+	827.402i	−0.188294	+	0.326135i	0.756388	+	0.654124i	$0.226962\pi$
$24$		0			0
$25$	1318.92	+	2284.44i	0.422055	+	0.731021i
$26$		0			0
$27$	−729.000			−0.192450
$28$		0			0
$29$	−7035.29			−1.55341			−0.776707	−	0.629862i	$-0.783111\pi$
$29$	−7035.29			−1.55341			−0.776707	+	0.629862i	$0.783111\pi$
$30$		0			0
$31$	630.596	+	1092.22i	0.117855	+	0.204130i	0.918917	−	0.394450i	$-0.129065\pi$
$31$	630.596	+	1092.22i	0.117855	+	0.204130i	−0.801063	+	0.598581i	$0.795732\pi$
$32$		0			0
$33$	−1875.19	+	3247.93i	−0.299751	+	0.519184i
$34$		0			0
$35$	891.689	−	2718.92i	0.123039	−	0.375168i
$36$		0			0
$37$	−4888.22	+	8466.65i	−0.587012	+	1.01673i	0.407610	+	0.913156i	$0.366362\pi$
$37$	−4888.22	+	8466.65i	−0.587012	+	1.01673i	−0.994621	+	0.103578i	$0.966971\pi$
$38$		0			0
$39$	3590.63	+	6219.16i	0.378015	+	0.654742i
$40$		0			0
$41$	−5400.95			−0.501777			−0.250888	−	0.968016i	$-0.580723\pi$
$41$	−5400.95			−0.501777			−0.250888	+	0.968016i	$0.580723\pi$
$42$		0			0
$43$	−19686.6			−1.62367			−0.811837	−	0.583885i	$-0.801532\pi$
$43$	−19686.6			−1.62367			−0.811837	+	0.583885i	$0.801532\pi$
$44$		0			0
$45$	−893.900	−	1548.28i	−0.0658048	−	0.113977i
$46$		0			0
$47$	1028.28	−	1781.04i	0.0678997	−	0.117606i	−0.830077	−	0.557649i	$-0.811704\pi$
$47$	1028.28	−	1781.04i	0.0678997	−	0.117606i	0.897977	+	0.440043i	$0.145037\pi$
$48$		0			0
$49$	15391.3	−	6751.63i	0.915765	−	0.401715i
$50$		0			0
$51$	−6189.98	+	10721.4i	−0.333245	+	0.577197i
$52$		0			0
$53$	−9011.37	−	15608.2i	−0.440658	−	0.763241i	0.557081	−	0.830458i	$-0.311921\pi$
$53$	−9011.37	−	15608.2i	−0.440658	−	0.763241i	−0.997738	+	0.0672170i	$0.978588\pi$
$54$		0			0
$55$	−9197.45			−0.409978
$56$		0			0
$57$	20817.2			0.848664
$58$		0			0
$59$	3717.84	+	6439.49i	0.139047	+	0.240836i	0.927136	−	0.374725i	$-0.122263\pi$
$59$	3717.84	+	6439.49i	0.139047	+	0.240836i	−0.788089	+	0.615561i	$0.788930\pi$
$60$		0			0
$61$	−1747.69	+	3027.09i	−0.0601368	+	0.104160i	−0.894526	−	0.447015i	$-0.852487\pi$
$61$	−1747.69	+	3027.09i	−0.0601368	+	0.104160i	0.834390	+	0.551175i	$0.185820\pi$
$62$		0			0
$63$	3272.39	−	9978.09i	0.103875	−	0.316735i
$64$		0			0
$65$	−8805.67	+	15251.9i	−0.258511	+	0.447754i
$66$		0			0
$67$	7928.21	+	13732.1i	0.215769	+	0.373722i	0.953510	−	0.301361i	$-0.0974410\pi$
$67$	7928.21	+	13732.1i	0.215769	+	0.373722i	−0.737741	+	0.675083i	$0.764108\pi$
$68$		0			0
$69$	−8598.62			−0.217423
$70$		0			0
$71$	−58133.5			−1.36861			−0.684306	−	0.729195i	$-0.739895\pi$
$71$	−58133.5			−1.36861			−0.684306	+	0.729195i	$0.739895\pi$
$72$		0			0
$73$	−19555.3	−	33870.8i	−0.429495	−	0.743907i	0.567334	−	0.823488i	$-0.307975\pi$
$73$	−19555.3	−	33870.8i	−0.429495	−	0.743907i	−0.996828	+	0.0795812i	$0.974642\pi$
$74$		0			0
$75$	−11870.3	+	20560.0i	−0.243674	+	0.422055i
$76$		0			0
$77$	−36038.1	−	40246.0i	−0.692684	−	0.773563i
$78$		0			0
$79$	4880.35	−	8453.01i	0.0879798	−	0.152385i	−0.818677	−	0.574254i	$-0.805292\pi$
$79$	4880.35	−	8453.01i	0.0879798	−	0.152385i	0.906657	+	0.421868i	$0.138626\pi$
$80$		0			0
$81$	−3280.50	−	5681.99i	−0.0555556	−	0.0962250i
$82$		0			0
$83$	70395.7			1.12163			0.560816	−	0.827940i	$-0.310487\pi$
$83$	70395.7			1.12163			0.560816	+	0.827940i	$0.310487\pi$
$84$		0			0
$85$	−30360.6			−0.455788
$86$		0			0
$87$	−31658.8	−	54834.7i	−0.448432	−	0.776707i
$88$		0			0
$89$	−72153.1	+	124973.i	−0.965562	+	1.67240i	−0.257464	+	0.966288i	$0.582887\pi$
$89$	−72153.1	+	124973.i	−0.965562	+	1.67240i	−0.708098	+	0.706115i	$0.750446\pi$
$90$		0			0
$91$	−101242.	+	21229.3i	−1.28161	+	0.268740i
$92$		0			0
$93$	−5675.36	+	9830.01i	−0.0680434	+	0.117855i
$94$		0			0
$95$	25526.1	+	44212.5i	0.290185	+	0.502616i
$96$		0			0
$97$	−79328.7			−0.856053			−0.428027	−	0.903766i	$-0.640791\pi$
$97$	−79328.7			−0.856053			−0.428027	+	0.903766i	$0.640791\pi$
$98$		0			0
$99$	−33753.5			−0.346123
$100$		0			0
$101$	42416.9	+	73468.2i	0.413747	+	0.716631i	0.995296	−	0.0968802i	$-0.0308864\pi$
$101$	42416.9	+	73468.2i	0.413747	+	0.716631i	−0.581549	+	0.813512i	$0.697553\pi$
$102$		0			0
$103$	10166.1	−	17608.3i	0.0944197	−	0.163540i	−0.814947	−	0.579536i	$-0.803234\pi$
$103$	10166.1	−	17608.3i	0.0944197	−	0.163540i	0.909366	+	0.415996i	$0.136567\pi$
$104$		0			0
$105$	25204.5	−	5285.11i	0.223102	−	0.0467822i
$106$		0			0
$107$	3481.09	−	6029.43i	0.0293938	−	0.0509116i	−0.850954	−	0.525240i	$-0.823976\pi$
$107$	3481.09	−	6029.43i	0.0293938	−	0.0509116i	0.880348	+	0.474328i	$0.157309\pi$
$108$		0			0
$109$	−56325.7	−	97559.0i	−0.454088	−	0.786504i	0.544547	−	0.838730i	$-0.316702\pi$
$109$	−56325.7	−	97559.0i	−0.454088	−	0.786504i	−0.998635	+	0.0522262i	$0.983368\pi$
$110$		0			0
$111$	−87988.0			−0.677823
$112$		0			0
$113$	112005.			0.825167			0.412583	−	0.910920i	$-0.364627\pi$
$113$	112005.			0.825167			0.412583	+	0.910920i	$0.364627\pi$
$114$		0			0
$115$	−10543.6	−	18262.1i	−0.0743439	−	0.128767i
$116$		0			0
$117$	−32315.7	+	55972.4i	−0.218247	+	0.378015i
$118$		0			0
$119$	−118961.	−	132851.i	−0.770083	−	0.860000i
$120$		0			0
$121$	−6297.91	+	10908.3i	−0.0391051	+	0.0677320i
$122$		0			0
$123$	−24304.3	−	42096.2i	−0.144850	−	0.250888i
$124$		0			0
$125$	−127195.			−0.728108
$126$		0			0
$127$	−82224.5			−0.452368			−0.226184	−	0.974085i	$-0.572625\pi$
$127$	−82224.5			−0.452368			−0.226184	+	0.974085i	$0.572625\pi$
$128$		0			0
$129$	−88589.5	−	153442.i	−0.468714	−	0.811837i
$130$		0			0
$131$	−87905.9	+	152258.i	−0.447548	+	0.775176i	−0.998226	−	0.0595417i	$-0.981036\pi$
$131$	−87905.9	+	152258.i	−0.447548	+	0.775176i	0.550677	+	0.834718i	$0.314369\pi$
$132$		0			0
$133$	−93445.8	+	284933.i	−0.458069	+	1.39673i
$134$		0			0
$135$	8045.10	−	13934.5i	0.0379924	−	0.0658048i
$136$		0			0
$137$	−15665.2	−	27132.9i	−0.0713072	−	0.123508i	0.828167	−	0.560481i	$-0.189384\pi$
$137$	−15665.2	−	27132.9i	−0.0713072	−	0.123508i	−0.899474	+	0.436973i	$0.856050\pi$
$138$		0			0
$139$	−152234.			−0.668305			−0.334152	−	0.942519i	$-0.608450\pi$
$139$	−152234.			−0.668305			−0.334152	+	0.942519i	$0.608450\pi$
$140$		0			0
$141$	18509.1			0.0784038
$142$		0			0
$143$	166250.	+	287953.i	0.679863	+	1.17756i
$144$		0			0
$145$	77640.1	−	134477.i	0.306666	−	0.531162i
$146$		0			0
$147$	121884.	+	89580.6i	0.465216	+	0.341917i
$148$		0			0
$149$	−181430.	+	314246.i	−0.669489	+	1.15959i	0.308558	+	0.951206i	$0.400154\pi$
$149$	−181430.	+	314246.i	−0.669489	+	1.15959i	−0.978047	+	0.208384i	$0.933180\pi$
$150$		0			0
$151$	102563.	+	177644.i	0.366056	+	0.634027i	0.988945	−	0.148283i	$-0.0473746\pi$
$151$	102563.	+	177644.i	0.366056	+	0.634027i	−0.622889	+	0.782310i	$0.714041\pi$
$152$		0			0
$153$	−111420.			−0.384798
$154$		0			0
$155$	−27836.5			−0.0930648
$156$		0			0
$157$	38636.0	+	66919.5i	0.125096	+	0.216672i	0.921770	−	0.387736i	$-0.126743\pi$
$157$	38636.0	+	66919.5i	0.125096	+	0.216672i	−0.796675	+	0.604409i	$0.793409\pi$
$158$		0			0
$159$	81102.3	−	140473.i	0.254414	−	0.440658i
$160$		0			0
$161$	38598.1	−	117692.i	0.117355	−	0.357836i
$162$		0			0
$163$	92465.7	−	160155.i	0.272591	−	0.472141i	−0.696934	−	0.717136i	$-0.745453\pi$
$163$	92465.7	−	160155.i	0.272591	−	0.472141i	0.969525	+	0.244994i	$0.0787861\pi$
$164$		0			0
$165$	−41388.5	−	71687.0i	−0.118350	−	0.204989i
$166$		0			0
$167$	−129262.			−0.358657			−0.179329	−	0.983789i	$-0.557393\pi$
$167$	−129262.			−0.358657			−0.179329	+	0.983789i	$0.557393\pi$
$168$		0			0
$169$	265380.			0.714746
$170$		0			0
$171$	93677.6	+	162254.i	0.244988	+	0.424332i
$172$		0			0
$173$	253934.	−	439826.i	0.645067	−	1.11729i	−0.339219	−	0.940707i	$-0.610163\pi$
$173$	253934.	−	439826.i	0.645067	−	1.11729i	0.984286	−	0.176582i	$-0.0565040\pi$
$174$		0			0
$175$	−228127.	−	254764.i	−0.563096	−	0.628844i
$176$		0			0
$177$	−33460.6	+	57955.4i	−0.0802786	+	0.139047i
$178$		0			0
$179$	−66294.5	−	114825.i	−0.154648	−	0.267859i	0.778283	−	0.627914i	$-0.216091\pi$
$179$	−66294.5	−	114825.i	−0.154648	−	0.267859i	−0.932931	+	0.360056i	$0.882758\pi$
$180$		0			0
$181$	−740060.			−1.67908			−0.839538	−	0.543301i	$-0.817174\pi$
$181$	−740060.			−1.67908			−0.839538	+	0.543301i	$0.817174\pi$
$182$		0			0
$183$	−31458.5			−0.0694400
$184$		0			0
$185$	−107891.	−	186873.i	−0.231769	−	0.401436i
$186$		0			0
$187$	−286603.	+	496410.i	−0.599344	+	1.03809i
$188$		0			0
$189$	92497.2	−	19395.7i	0.188354	−	0.0394958i
$190$		0			0
$191$	−291366.	+	504661.i	−0.577904	+	1.00096i	0.417815	+	0.908532i	$0.362796\pi$
$191$	−291366.	+	504661.i	−0.577904	+	1.00096i	−0.995719	+	0.0924273i	$0.970537\pi$
$192$		0			0
$193$	200452.	+	347193.i	0.387362	+	0.670931i	0.992094	−	0.125498i	$-0.0400529\pi$
$193$	200452.	+	347193.i	0.387362	+	0.670931i	−0.604731	+	0.796429i	$0.706720\pi$
$194$		0			0
$195$	−158502.			−0.298503
$196$		0			0
$197$	671589.			1.23293			0.616464	−	0.787383i	$-0.288564\pi$
$197$	671589.			1.23293			0.616464	+	0.787383i	$0.288564\pi$
$198$		0			0
$199$	−227511.	−	394060.i	−0.407258	−	0.705391i	0.587324	−	0.809352i	$-0.300182\pi$
$199$	−227511.	−	394060.i	−0.407258	−	0.705391i	−0.994581	+	0.103961i	$0.966848\pi$
$200$		0			0
$201$	−71353.9	+	123589.i	−0.124574	+	0.215769i
$202$		0			0
$203$	892654.	−	187180.i	1.52035	−	0.318801i
$204$		0			0
$205$	59603.8	−	103237.i	0.0990580	−	0.171573i
$206$		0			0
$207$	−38693.8	−	67019.6i	−0.0627647	−	0.108712i
$208$		0			0
$209$	963860.			1.52633
$210$		0			0
$211$	1.19545e6			1.84852			0.924260	−	0.381764i	$-0.124683\pi$
$211$	1.19545e6			1.84852			0.924260	+	0.381764i	$0.124683\pi$
$212$		0			0
$213$	−261601.	−	453105.i	−0.395084	−	0.684306i
$214$		0			0
$215$	217257.	−	376300.i	0.320537	−	0.555186i
$216$		0			0
$217$	−109071.	−	121806.i	−0.157239	−	0.175598i
$218$		0			0
$219$	175998.	−	304837.i	0.247969	−	0.429495i
$220$		0			0
$221$	548788.	+	950529.i	0.755830	+	1.30914i
$222$		0			0
$223$	−296529.			−0.399305			−0.199653	−	0.979867i	$-0.563981\pi$
$223$	−296529.			−0.399305			−0.199653	+	0.979867i	$0.563981\pi$
$224$		0			0
$225$	−213665.			−0.281370
$226$		0			0
$227$	109073.	+	188920.i	0.140492	+	0.243340i	0.927682	−	0.373371i	$-0.121798\pi$
$227$	109073.	+	188920.i	0.140492	+	0.243340i	−0.787190	+	0.616711i	$0.788465\pi$
$228$		0			0
$229$	614602.	−	1.06452e6i	0.774471	−	1.34142i	−0.160620	−	0.987016i	$-0.551349\pi$
$229$	614602.	−	1.06452e6i	0.774471	−	1.34142i	0.935091	−	0.354407i	$-0.115317\pi$
$230$		0			0
$231$	151515.	−	461996.i	0.186821	−	0.569650i
$232$		0			0
$233$	31472.1	−	54511.3i	0.0379784	−	0.0657805i	−0.846411	−	0.532530i	$-0.821242\pi$
$233$	31472.1	−	54511.3i	0.0379784	−	0.0657805i	0.884390	+	0.466749i	$0.154575\pi$
$234$		0			0
$235$	22695.8	+	39310.4i	0.0268088	+	0.0464341i
$236$		0			0
$237$	87846.2			0.101590
$238$		0			0
$239$	−219330.			−0.248372			−0.124186	−	0.992259i	$-0.539632\pi$
$239$	−219330.			−0.248372			−0.124186	+	0.992259i	$0.539632\pi$
$240$		0			0
$241$	−216932.	−	375737.i	−0.240592	−	0.416717i	0.720291	−	0.693672i	$-0.244008\pi$
$241$	−216932.	−	375737.i	−0.240592	−	0.416717i	−0.960883	+	0.276955i	$0.910675\pi$
$242$		0			0
$243$	29524.5	−	51137.9i	0.0320750	−	0.0555556i
$244$		0			0
$245$	−40800.4	+	368707.i	−0.0434259	+	0.392433i
$246$		0			0
$247$	922803.	−	1.59834e6i	0.962424	−	1.66697i
$248$		0			0
$249$	316780.	+	548680.i	0.323787	+	0.560816i
$250$		0			0
$251$	1.71109e6			1.71431			0.857155	−	0.515059i	$-0.172230\pi$
$251$	1.71109e6			1.71431			0.857155	+	0.515059i	$0.172230\pi$
$252$		0			0
$253$	−398125.			−0.391037
$254$		0			0
$255$	−136623.	−	236638.i	−0.131575	−	0.227894i
$256$		0			0
$257$	−492029.	+	852219.i	−0.464684	+	0.804856i	−0.999187	−	0.0403103i	$-0.987165\pi$
$257$	−492029.	+	852219.i	−0.464684	+	0.804856i	0.534503	+	0.845166i	$0.320499\pi$
$258$		0			0
$259$	394967.	−	1.20432e6i	0.365857	−	1.11556i
$260$		0			0
$261$	284929.	−	493512.i	0.258902	−	0.448432i
$262$		0			0
$263$	273548.	+	473799.i	0.243862	+	0.422381i	0.961811	−	0.273714i	$-0.0882523\pi$
$263$	273548.	+	473799.i	0.243862	+	0.422381i	−0.717949	+	0.696096i	$0.754919\pi$
$264$		0			0
$265$	397791.			0.347969
$266$		0			0
$267$	−1.29876e6			−1.11493
$268$		0			0
$269$	320556.	+	555219.i	0.270099	+	0.467825i	0.968887	−	0.247504i	$-0.0796102\pi$
$269$	320556.	+	555219.i	0.270099	+	0.467825i	−0.698788	+	0.715329i	$0.746277\pi$
$270$		0			0
$271$	−274006.	+	474593.i	−0.226640	+	0.392552i	−0.956810	−	0.290713i	$-0.906107\pi$
$271$	−274006.	+	474593.i	−0.226640	+	0.392552i	0.730170	+	0.683265i	$0.239441\pi$
$272$		0			0
$273$	−621054.	−	693569.i	−0.504339	−	0.563227i
$274$		0			0
$275$	−549607.	+	951948.i	−0.438249	+	0.759069i
$276$		0			0
$277$	−837070.	−	1.44985e6i	−0.655485	−	1.13533i	−0.981772	−	0.190062i	$-0.939131\pi$
$277$	−837070.	−	1.44985e6i	−0.655485	−	1.13533i	0.326287	−	0.945271i	$-0.394202\pi$
$278$		0			0
$279$	−102157.			−0.0785698
$280$		0			0
$281$	1.81078e6			1.36804			0.684021	−	0.729462i	$-0.260230\pi$
$281$	1.81078e6			1.36804			0.684021	+	0.729462i	$0.260230\pi$
$282$		0			0
$283$	1.25657e6	+	2.17645e6i	0.932657	+	1.61541i	0.778759	+	0.627323i	$0.215849\pi$
$283$	1.25657e6	+	2.17645e6i	0.932657	+	1.61541i	0.153898	+	0.988087i	$0.450817\pi$
$284$		0			0
$285$	−229735.	+	397913.i	−0.167539	+	0.290185i
$286$		0			0
$287$	685285.	−	143697.i	0.491096	−	0.102978i
$288$		0			0
$289$	−236142.	+	409009.i	−0.166314	+	0.288064i
$290$		0			0
$291$	−356979.	−	618306.i	−0.247121	−	0.428027i
$292$		0			0
$293$	107228.			0.0729691			0.0364845	−	0.999334i	$-0.488384\pi$
$293$	107228.			0.0729691			0.0364845	+	0.999334i	$0.488384\pi$
$294$		0			0
$295$	−164117.			−0.109799
$296$		0			0
$297$	−151891.	−	263082.i	−0.0999171	−	0.173061i
$298$		0			0
$299$	−381166.	+	660199.i	−0.246568	+	0.427068i
$300$		0			0
$301$	2.49788e6	−	523778.i	1.58911	−	0.333220i
$302$		0			0
$303$	−381752.	+	661214.i	−0.238877	+	0.413747i
$304$		0			0
$305$	−38574.4	−	66812.8i	−0.0237437	−	0.0411254i
$306$		0			0
$307$	−1.49622e6			−0.906042			−0.453021	−	0.891500i	$-0.649654\pi$
$307$	−1.49622e6			−0.906042			−0.453021	+	0.891500i	$0.649654\pi$
$308$		0			0
$309$	182990.			0.109027
$310$		0			0
$311$	430601.	+	745823.i	0.252449	+	0.437255i	0.964200	−	0.265178i	$-0.0854306\pi$
$311$	430601.	+	745823.i	0.252449	+	0.437255i	−0.711750	+	0.702433i	$0.752097\pi$
$312$		0			0
$313$	251969.	−	436422.i	0.145374	−	0.251794i	−0.784139	−	0.620586i	$-0.786895\pi$
$313$	251969.	−	436422.i	0.145374	−	0.251794i	0.929512	+	0.368791i	$0.120228\pi$
$314$		0			0
$315$	154613.	+	172666.i	0.0877952	+	0.0980464i
$316$		0			0
$317$	240004.	−	415700.i	0.134144	−	0.232344i	−0.791126	−	0.611653i	$-0.790505\pi$
$317$	240004.	−	415700.i	0.134144	−	0.232344i	0.925270	+	0.379309i	$0.123838\pi$
$318$		0			0
$319$	−1.46584e6	−	2.53890e6i	−0.806508	−	1.39691i
$320$		0			0
$321$	62659.7			0.0339411
$322$		0			0
$323$	3.18168e6			1.69688
$324$		0			0
$325$	1.05239e6	+	1.82280e6i	0.552674	+	0.957259i
$326$		0			0
$327$	506931.	−	878031.i	0.262168	−	0.454088i
$328$		0			0
$329$	−83084.8	+	253341.i	−0.0423187	+	0.129037i
$330$		0			0
$331$	−1.09961e6	+	1.90458e6i	−0.551656	+	0.955497i	0.446499	+	0.894784i	$0.352671\pi$
$331$	−1.09961e6	+	1.90458e6i	−0.551656	+	0.955497i	−0.998155	+	0.0607125i	$0.980663\pi$
$332$		0			0
$333$	−395946.	−	685799.i	−0.195671	−	0.338911i
$334$		0			0
$335$	−349977.			−0.170383
$336$		0			0
$337$	−1.35725e6			−0.651008			−0.325504	−	0.945541i	$-0.605534\pi$
$337$	−1.35725e6			−0.651008			−0.325504	+	0.945541i	$0.605534\pi$
$338$		0			0
$339$	504023.	+	872993.i	0.238205	+	0.412583i
$340$		0			0
$341$	−262775.	+	455140.i	−0.122377	+	0.211963i
$342$		0			0
$343$	−1.77325e6	+	1.26616e6i	−0.813830	+	0.581103i
$344$		0			0
$345$	94892.6	−	164359.i	0.0429225	−	0.0743439i
$346$		0			0
$347$	1.89970e6	+	3.29038e6i	0.846959	+	1.46698i	0.883909	+	0.467658i	$0.154902\pi$
$347$	1.89970e6	+	3.29038e6i	0.846959	+	1.46698i	−0.0369507	+	0.999317i	$0.511764\pi$
$348$		0			0
$349$	−1.31753e6			−0.579024			−0.289512	−	0.957174i	$-0.593493\pi$
$349$	−1.31753e6			−0.579024			−0.289512	+	0.957174i	$0.593493\pi$
$350$		0			0
$351$	−581682.			−0.252010
$352$		0			0
$353$	1.71253e6	+	2.96618e6i	0.731477	+	1.26695i	0.956252	+	0.292544i	$0.0945018\pi$
$353$	1.71253e6	+	2.96618e6i	0.731477	+	1.26695i	−0.224775	+	0.974411i	$0.572165\pi$
$354$		0			0
$355$	641549.	−	1.11120e6i	0.270184	−	0.467972i
$356$		0			0
$357$	500148.	−	1.52504e6i	0.207696	−	0.633302i
$358$		0			0
$359$	735419.	−	1.27378e6i	0.301161	−	0.521626i	−0.675238	−	0.737600i	$-0.735959\pi$
$359$	735419.	−	1.27378e6i	0.301161	−	0.521626i	0.976399	+	0.215974i	$0.0692925\pi$
$360$		0			0
$361$	−1.43700e6	−	2.48895e6i	−0.580347	−	1.00519i
$362$		0			0
$363$	−113362.			−0.0451547
$364$		0			0
$365$	863235.			0.339154
$366$		0			0
$367$	−2.49039e6	−	4.31349e6i	−0.965168	−	1.67172i	−0.709163	−	0.705045i	$-0.750927\pi$
$367$	−2.49039e6	−	4.31349e6i	−0.965168	−	1.67172i	−0.256006	−	0.966675i	$-0.582407\pi$
$368$		0			0
$369$	218738.	−	378866.i	0.0836295	−	0.144850i
$370$		0			0
$371$	1.55865e6	+	1.74064e6i	0.587915	+	0.656561i
$372$		0			0
$373$	−1.98024e6	+	3.42987e6i	−0.736962	+	1.27646i	0.216895	+	0.976195i	$0.430407\pi$
$373$	−1.98024e6	+	3.42987e6i	−0.736962	+	1.27646i	−0.953857	+	0.300261i	$0.902926\pi$
$374$		0			0
$375$	−572379.	−	991389.i	−0.210187	−	0.364054i
$376$		0			0
$377$	−5.61359e6			−2.03417
$378$		0			0
$379$	1.75155e6			0.626359			0.313179	−	0.949694i	$-0.398606\pi$
$379$	1.75155e6			0.626359			0.313179	+	0.949694i	$0.398606\pi$
$380$		0			0
$381$	−370010.	−	640876.i	−0.130587	−	0.226184i
$382$		0			0
$383$	1.56917e6	−	2.71788e6i	0.546604	−	0.946745i	−0.451900	−	0.892068i	$-0.649254\pi$
$383$	1.56917e6	−	2.71788e6i	0.546604	−	0.946745i	0.998504	−	0.0546771i	$-0.0174129\pi$
$384$		0			0
$385$	1.16699e6	−	244706.i	0.401252	−	0.0841381i
$386$		0			0
$387$	797306.	−	1.38097e6i	0.270612	−	0.468714i
$388$		0			0
$389$	526262.	+	911512.i	0.176331	+	0.305414i	0.940621	−	0.339459i	$-0.110244\pi$
$389$	526262.	+	911512.i	0.176331	+	0.305414i	−0.764290	+	0.644872i	$0.776911\pi$
$390$		0			0
$391$	−1.31420e6			−0.434731
$392$		0			0
$393$	−1.58231e6			−0.516784
$394$		0			0
$395$	107717.	+	186571.i	0.0347370	+	0.0601662i
$396$		0			0
$397$	−227362.	+	393803.i	−0.0724005	+	0.125401i	−0.899953	−	0.435987i	$-0.856399\pi$
$397$	−227362.	+	393803.i	−0.0724005	+	0.125401i	0.827552	+	0.561389i	$0.189733\pi$
$398$		0			0
$399$	−2.64134e6	+	553861.i	−0.830600	+	0.174168i
$400$		0			0
$401$	1.44216e6	−	2.49789e6i	0.447870	−	0.775733i	−0.550378	−	0.834916i	$-0.685516\pi$
$401$	1.44216e6	−	2.49789e6i	0.447870	−	0.775733i	0.998247	+	0.0591831i	$0.0188496\pi$
$402$		0			0
$403$	503164.	+	871505.i	0.154329	+	0.267305i
$404$		0			0
$405$	144812.			0.0438699
$406$		0			0
$407$	−4.07394e6			−1.21907
$408$		0			0
$409$	112957.	+	195647.i	0.0333891	+	0.0578317i	0.882237	−	0.470805i	$-0.156037\pi$
$409$	112957.	+	195647.i	0.0333891	+	0.0578317i	−0.848848	+	0.528637i	$0.822703\pi$
$410$		0			0
$411$	140986.	−	244196.i	0.0411692	−	0.0713072i
$412$		0			0
$413$	−643056.	−	718141.i	−0.185513	−	0.207174i
$414$		0			0
$415$	−776873.	+	1.34558e6i	−0.221426	+	0.383522i
$416$		0			0
$417$	−685053.	−	1.18655e6i	−0.192923	−	0.334152i
$418$		0			0
$419$	−4.31027e6			−1.19941			−0.599707	−	0.800220i	$-0.704716\pi$
$419$	−4.31027e6			−1.19941			−0.599707	+	0.800220i	$0.704716\pi$
$420$		0			0
$421$	1.25088e6			0.343962			0.171981	−	0.985100i	$-0.444983\pi$
$421$	1.25088e6			0.343962			0.171981	+	0.985100i	$0.444983\pi$
$422$		0			0
$423$	83290.9	+	144264.i	0.0226332	+	0.0392019i
$424$		0			0
$425$	−1.81424e6	+	3.14236e6i	−0.487218	+	0.843887i
$426$		0			0
$427$	141213.	−	430583.i	0.0374804	−	0.114285i
$428$		0			0
$429$	−1.49625e6	+	2.59158e6i	−0.392519	+	0.679863i
$430$		0			0
$431$	−2.20397e6	−	3.81738e6i	−0.571494	−	0.989857i	−0.996413	−	0.0846252i	$-0.973031\pi$
$431$	−2.20397e6	−	3.81738e6i	−0.571494	−	0.989857i	0.424919	−	0.905231i	$-0.360303\pi$
$432$		0			0
$433$	−1.60951e6			−0.412549			−0.206274	−	0.978494i	$-0.566134\pi$
$433$	−1.60951e6			−0.412549			−0.206274	+	0.978494i	$0.566134\pi$
$434$		0			0
$435$	1.39752e6			0.354108
$436$		0			0
$437$	1.10493e6	+	1.91380e6i	0.276779	+	0.479395i
$438$		0			0
$439$	−2.21226e6	+	3.83175e6i	−0.547867	+	0.948933i	0.450554	+	0.892749i	$0.351227\pi$
$439$	−2.21226e6	+	3.83175e6i	−0.547867	+	0.948933i	−0.998420	+	0.0561836i	$0.982107\pi$
$440$		0			0
$441$	−149732.	+	1.35311e6i	−0.0366622	+	0.331311i
$442$		0			0
$443$	−1.80719e6	+	3.13015e6i	−0.437517	+	0.757801i	−0.997497	−	0.0707044i	$-0.977475\pi$
$443$	−1.80719e6	+	3.13015e6i	−0.437517	+	0.757801i	0.559980	+	0.828506i	$0.310809\pi$
$444$		0			0
$445$	−1.59254e6	−	2.75835e6i	−0.381232	−	0.660313i
$446$		0			0
$447$	−3.26574e6			−0.773060
$448$		0			0
$449$	−467024.			−0.109326			−0.0546630	−	0.998505i	$-0.517408\pi$
$449$	−467024.			−0.109326			−0.0546630	+	0.998505i	$0.517408\pi$
$450$		0			0
$451$	−1.12531e6	−	1.94910e6i	−0.260515	−	0.451225i
$452$		0			0
$453$	−923065.	+	1.59880e6i	−0.211342	+	0.366056i
$454$		0			0
$455$	711495.	−	2.16947e6i	0.161118	−	0.491276i
$456$		0			0
$457$	300626.	−	520699.i	0.0673342	−	0.116626i	−0.830393	−	0.557178i	$-0.811884\pi$
$457$	300626.	−	520699.i	0.0673342	−	0.116626i	0.897727	+	0.440552i	$0.145217\pi$
$458$		0			0
$459$	−501388.	−	868430.i	−0.111082	−	0.192399i
$460$		0			0
$461$	2.87193e6			0.629392			0.314696	−	0.949192i	$-0.398097\pi$
$461$	2.87193e6			0.629392			0.314696	+	0.949192i	$0.398097\pi$
$462$		0			0
$463$	2.91502e6			0.631959			0.315979	−	0.948766i	$-0.397667\pi$
$463$	2.91502e6			0.631959			0.315979	+	0.948766i	$0.397667\pi$
$464$		0			0
$465$	−125264.	−	216964.i	−0.0268655	−	0.0465324i
$466$		0			0
$467$	−3.59688e6	+	6.22998e6i	−0.763192	+	1.32189i	0.178006	+	0.984029i	$0.443035\pi$
$467$	−3.59688e6	+	6.22998e6i	−0.763192	+	1.32189i	−0.941197	+	0.337857i	$0.890298\pi$
$468$		0			0
$469$	−1.37130e6	−	1.53142e6i	−0.287873	−	0.321486i
$470$		0			0
$471$	−347724.	+	602275.i	−0.0722241	+	0.125096i
$472$		0			0
$473$	−4.10179e6	−	7.10451e6i	−0.842986	−	1.46009i
$474$		0			0
$475$	6.10140e6			1.24078
$476$		0			0
$477$	1.45984e6			0.293772
$478$		0			0
$479$	−1.39824e6	−	2.42183e6i	−0.278448	−	0.482286i	0.692551	−	0.721369i	$-0.256487\pi$
$479$	−1.39824e6	−	2.42183e6i	−0.278448	−	0.482286i	−0.970999	+	0.239083i	$0.923153\pi$
$480$		0			0
$481$	−3.90040e6	+	6.75569e6i	−0.768682	+	1.33140i
$482$		0			0
$483$	1.09101e6	−	228774.i	0.212795	−	0.0446209i
$484$		0			0
$485$	875456.	−	1.51633e6i	0.168997	−	0.292712i
$486$		0			0
$487$	−1.46624e6	−	2.53960e6i	−0.280144	−	0.485224i	0.691276	−	0.722591i	$-0.257049\pi$
$487$	−1.46624e6	−	2.53960e6i	−0.280144	−	0.485224i	−0.971420	+	0.237367i	$0.923716\pi$
$488$		0			0
$489$	1.66438e6			0.314761
$490$		0			0
$491$	3.06121e6			0.573046			0.286523	−	0.958073i	$-0.407501\pi$
$491$	3.06121e6			0.573046			0.286523	+	0.958073i	$0.407501\pi$
$492$		0			0
$493$	−4.83870e6	−	8.38087e6i	−0.896626	−	1.55300i
$494$		0			0
$495$	372497.	−	645183.i	0.0683297	−	0.118350i
$496$		0			0
$497$	7.37611e6	−	1.54669e6i	1.33948	−	0.280875i
$498$		0			0
$499$	−3.27577e6	+	5.67380e6i	−0.588928	+	1.02005i	0.405445	+	0.914119i	$0.367116\pi$
$499$	−3.27577e6	+	5.67380e6i	−0.588928	+	1.02005i	−0.994373	+	0.105934i	$0.966217\pi$
$500$		0			0
$501$	−581679.	−	1.00750e6i	−0.103535	−	0.179329i
$502$		0			0
$503$	1.58524e6			0.279367			0.139684	−	0.990196i	$-0.455391\pi$
$503$	1.58524e6			0.279367			0.139684	+	0.990196i	$0.455391\pi$
$504$		0			0
$505$	−1.87242e6			−0.326719
$506$		0			0
$507$	1.19421e6	+	2.06843e6i	0.206329	+	0.357373i
$508$		0			0
$509$	−3.38991e6	+	5.87149e6i	−0.579953	+	1.00451i	0.415530	+	0.909579i	$0.363596\pi$
$509$	−3.38991e6	+	5.87149e6i	−0.579953	+	1.00451i	−0.995484	+	0.0949297i	$0.969737\pi$
$510$		0			0
$511$	3.38239e6	+	3.77732e6i	0.573022	+	0.639929i
$512$		0			0
$513$	−843098.	+	1.46029e6i	−0.141444	+	0.244988i
$514$		0			0
$515$	224383.	+	388643.i	0.0372796	+	0.0645702i
$516$		0			0
$517$	856990.			0.141010
$518$		0			0
$519$	4.57081e6			0.744859
$520$		0			0
$521$	5.06918e6	+	8.78008e6i	0.818170	+	1.41711i	0.907029	+	0.421067i	$0.138344\pi$
$521$	5.06918e6	+	8.78008e6i	0.818170	+	1.41711i	−0.0888599	+	0.996044i	$0.528322\pi$
$522$		0			0
$523$	49998.6	−	86600.2i	0.00799289	−	0.0138441i	−0.862001	−	0.506906i	$-0.830789\pi$
$523$	49998.6	−	86600.2i	0.00799289	−	0.0138441i	0.869994	+	0.493062i	$0.164122\pi$
$524$		0			0
$525$	959116.	−	2.92452e6i	0.151870	−	0.463080i
$526$		0			0
$527$	−867416.	+	1.50241e6i	−0.136051	+	0.235647i
$528$		0			0
$529$	2.76178e6	+	4.78354e6i	0.429091	+	0.743207i
$530$		0			0
$531$	−602290.			−0.0926978
$532$		0			0
$533$	−4.30952e6			−0.657068
$534$		0			0
$535$	76833.3	+	133079.i	0.0116055	+	0.0201014i
$536$		0			0
$537$	596651.	−	1.03343e6i	0.0892862	−	0.154648i
$538$		0			0
$539$	5.64338e6	+	4.14768e6i	0.836695	+	0.614941i
$540$		0			0
$541$	59871.4	−	103700.i	0.00879481	−	0.0152331i	−0.861595	−	0.507597i	$-0.830534\pi$
$541$	59871.4	−	103700.i	0.00879481	−	0.0152331i	0.870389	+	0.492364i	$0.163867\pi$
$542$		0			0
$543$	−3.33027e6	−	5.76820e6i	−0.484708	−	0.839538i
$544$		0			0
$545$	2.48640e6			0.358574
$546$		0			0
$547$	−236568.			−0.0338056			−0.0169028	−	0.999857i	$-0.505381\pi$
$547$	−236568.			−0.0338056			−0.0169028	+	0.999857i	$0.505381\pi$
$548$		0			0
$549$	−141563.	−	245194.i	−0.0200456	−	0.0347200i
$550$		0			0
$551$	−8.13641e6	+	1.40927e7i	−1.14170	+	1.97749i
$552$		0			0
$553$	−394330.	+	1.20238e6i	−0.0548336	+	0.167198i
$554$		0			0
$555$	971018.	−	1.68185e6i	0.133812	−	0.231769i
$556$		0			0
$557$	2.41833e6	+	4.18867e6i	0.330277	+	0.572056i	0.982566	−	0.185914i	$-0.0595247\pi$
$557$	2.41833e6	+	4.18867e6i	0.330277	+	0.572056i	−0.652289	+	0.757970i	$0.726191\pi$
$558$		0			0
$559$	−1.57083e7			−2.12617
$560$		0			0
$561$	−5.15885e6			−0.692063
$562$		0			0
$563$	−2.74147e6	−	4.74837e6i	−0.364513	−	0.631355i	0.624185	−	0.781277i	$-0.285431\pi$
$563$	−2.74147e6	−	4.74837e6i	−0.364513	−	0.631355i	−0.988698	+	0.149922i	$0.952098\pi$
$564$		0			0
$565$	−1.23607e6	+	2.14093e6i	−0.162900	+	0.282151i
$566$		0			0
$567$	567412.	+	633664.i	0.0741209	+	0.0827754i
$568$		0			0
$569$	3.48830e6	−	6.04191e6i	0.451682	−	0.782336i	−0.546809	−	0.837258i	$-0.684157\pi$
$569$	3.48830e6	−	6.04191e6i	0.451682	−	0.782336i	0.998491	+	0.0549213i	$0.0174908\pi$
$570$		0			0
$571$	−5.88573e6	−	1.01944e7i	−0.755458	−	1.30849i	−0.945146	−	0.326647i	$-0.894081\pi$
$571$	−5.88573e6	−	1.01944e7i	−0.755458	−	1.30849i	0.189689	−	0.981844i	$-0.439252\pi$
$572$		0			0
$573$	−5.24459e6			−0.667306
$574$		0			0
$575$	−2.52020e6			−0.317882
$576$		0			0
$577$	−3.38646e6	−	5.86552e6i	−0.423454	−	0.733444i	0.572820	−	0.819681i	$-0.305849\pi$
$577$	−3.38646e6	−	5.86552e6i	−0.423454	−	0.733444i	−0.996275	+	0.0862365i	$0.972516\pi$
$578$		0			0
$579$	−1.80407e6	+	3.12474e6i	−0.223644	+	0.387362i
$580$		0			0
$581$	−8.93197e6	+	1.87294e6i	−1.09776	+	0.230188i
$582$		0			0
$583$	3.75512e6	−	6.50407e6i	0.457565	−	0.792526i
$584$		0			0
$585$	−713259.	−	1.23540e6i	−0.0861703	−	0.149251i
$586$		0			0
$587$	1.05020e7			1.25799			0.628996	−	0.777408i	$-0.283466\pi$
$587$	1.05020e7			1.25799			0.628996	+	0.777408i	$0.283466\pi$
$588$		0			0
$589$	2.91717e6			0.346476
$590$		0			0
$591$	3.02215e6	+	5.23452e6i	0.355916	+	0.616464i
$592$		0			0
$593$	3.79537e6	−	6.57378e6i	0.443218	−	0.767676i	−0.554708	−	0.832045i	$-0.687170\pi$
$593$	3.79537e6	−	6.57378e6i	0.443218	−	0.767676i	0.997926	+	0.0643687i	$0.0205034\pi$
$594$		0			0
$595$	3.85223e6	−	807770.i	0.446087	−	0.0935396i
$596$		0			0
$597$	2.04760e6	−	3.54654e6i	0.235130	−	0.407258i
$598$		0			0
$599$	−6.50992e6	−	1.12755e7i	−0.741325	−	1.28401i	−0.951892	−	0.306434i	$-0.900864\pi$
$599$	−6.50992e6	−	1.12755e7i	−0.741325	−	1.28401i	0.210567	−	0.977580i	$-0.432469\pi$
$600$		0			0
$601$	1.41821e7			1.60160			0.800801	−	0.598931i	$-0.204408\pi$
$601$	1.41821e7			1.60160			0.800801	+	0.598931i	$0.204408\pi$
$602$		0			0
$603$	−1.28437e6			−0.143846
$604$		0			0
$605$	−139005.	−	240764.i	−0.0154398	−	0.0267426i
$606$		0			0
$607$	6.03862e6	−	1.04592e7i	0.665221	−	1.15220i	−0.314004	−	0.949422i	$-0.601671\pi$
$607$	6.03862e6	−	1.04592e7i	0.665221	−	1.15220i	0.979225	−	0.202775i	$-0.0649961\pi$
$608$		0			0
$609$	5.47587e6	+	6.11524e6i	0.598287	+	0.668144i
$610$		0			0
$611$	820485.	−	1.42112e6i	0.0889135	−	0.154003i
$612$		0			0
$613$	1.36265e6	+	2.36018e6i	0.146465	+	0.253684i	0.929918	−	0.367766i	$-0.119877\pi$
$613$	1.36265e6	+	2.36018e6i	0.146465	+	0.253684i	−0.783454	+	0.621450i	$0.786544\pi$
$614$		0			0
$615$	1.07287e6			0.114382
$616$		0			0
$617$	8.47094e6			0.895816			0.447908	−	0.894080i	$-0.352169\pi$
$617$	8.47094e6			0.895816			0.447908	+	0.894080i	$0.352169\pi$
$618$		0			0
$619$	8.69174e6	+	1.50545e7i	0.911759	+	1.57921i	0.811578	+	0.584243i	$0.198609\pi$
$619$	8.69174e6	+	1.50545e7i	0.911759	+	1.57921i	0.100180	+	0.994969i	$0.468058\pi$
$620$		0			0
$621$	348244.	−	603176.i	0.0362372	−	0.0627647i
$622$		0			0
$623$	5.82995e6	−	1.77765e7i	0.601789	−	1.83496i
$624$		0			0
$625$	−2.71793e6	+	4.70759e6i	−0.278316	+	0.482058i
$626$		0			0
$627$	4.33737e6	+	7.51255e6i	0.440613	+	0.763165i
$628$		0			0
$629$	−1.34480e7			−1.35529
$630$		0			0
$631$	6.45149e6			0.645040			0.322520	−	0.946563i	$-0.395470\pi$
$631$	6.45149e6			0.645040			0.322520	+	0.946563i	$0.395470\pi$
$632$		0			0
$633$	5.37951e6	+	9.31759e6i	0.533622	+	0.924260i
$634$		0			0
$635$	907413.	−	1.57169e6i	0.0893040	−	0.154679i
$636$		0			0
$637$	1.22810e7	−	5.38725e6i	1.19918	−	0.526039i
$638$		0			0
$639$	2.35440e6	−	4.07795e6i	0.228102	−	0.395084i
$640$		0			0
$641$	−8.88581e6	−	1.53907e7i	−0.854185	−	1.47949i	−0.877399	−	0.479761i	$-0.840723\pi$
$641$	−8.88581e6	−	1.53907e7i	−0.854185	−	1.47949i	0.0232142	−	0.999731i	$-0.492610\pi$
$642$		0			0
$643$	9.34806e6			0.891649			0.445825	−	0.895120i	$-0.352910\pi$
$643$	9.34806e6			0.891649			0.445825	+	0.895120i	$0.352910\pi$
$644$		0			0
$645$	3.91063e6			0.370124
$646$		0			0
$647$	2.67193e6	+	4.62792e6i	0.250937	+	0.434635i	0.963784	−	0.266684i	$-0.0859280\pi$
$647$	2.67193e6	+	4.62792e6i	0.250937	+	0.434635i	−0.712847	+	0.701319i	$0.752595\pi$
$648$		0			0
$649$	−1.54926e6	+	2.68340e6i	−0.144382	+	0.250077i
$650$		0			0
$651$	458567.	−	1.39825e6i	0.0424083	−	0.129310i
$652$		0			0
$653$	598506.	−	1.03664e6i	0.0549270	−	0.0951363i	−0.837255	−	0.546813i	$-0.815841\pi$
$653$	598506.	−	1.03664e6i	0.0549270	−	0.0951363i	0.892182	+	0.451677i	$0.149174\pi$
$654$		0			0
$655$	−1.94022e6	−	3.36057e6i	−0.176705	−	0.306062i
$656$		0			0
$657$	3.16796e6			0.286330
$658$		0			0
$659$	1.17541e7			1.05433			0.527163	−	0.849764i	$-0.323256\pi$
$659$	1.17541e7			1.05433			0.527163	+	0.849764i	$0.323256\pi$
$660$		0			0
$661$	9.00573e6	+	1.55984e7i	0.801706	+	1.38860i	0.918493	+	0.395438i	$0.129407\pi$
$661$	9.00573e6	+	1.55984e7i	0.801706	+	1.38860i	−0.116787	+	0.993157i	$0.537259\pi$
$662$		0			0
$663$	−4.93910e6	+	8.55476e6i	−0.436379	+	0.755830i
$664$		0			0
$665$	−4.41512e6	−	4.93064e6i	−0.387158	−	0.432364i
$666$		0			0
$667$	3.36076e6	−	5.82102e6i	0.292498	−	0.506622i
$668$		0			0
$669$	−1.33438e6	−	2.31122e6i	−0.115270	−	0.199653i
$670$		0			0
$671$	−1.45656e6			−0.124888
$672$		0			0
$673$	1.40977e7			1.19981			0.599904	−	0.800072i	$-0.295206\pi$
$673$	1.40977e7			1.19981			0.599904	+	0.800072i	$0.295206\pi$
$674$		0			0
$675$	−961494.	−	1.66536e6i	−0.0812245	−	0.140685i
$676$		0			0
$677$	−3.82793e6	+	6.63018e6i	−0.320991	+	0.555973i	−0.980693	−	0.195555i	$-0.937349\pi$
$677$	−3.82793e6	+	6.63018e6i	−0.320991	+	0.555973i	0.659702	+	0.751527i	$0.270683\pi$
$678$		0			0
$679$	1.00654e7	−	2.11061e6i	0.837832	−	0.175684i
$680$		0			0
$681$	−981657.	+	1.70028e6i	−0.0811133	+	0.140492i
$682$		0			0
$683$	−5.43378e6	−	9.41158e6i	−0.445708	−	0.771988i	0.552394	−	0.833583i	$-0.313715\pi$
$683$	−5.43378e6	−	9.41158e6i	−0.445708	−	0.771988i	−0.998101	+	0.0615953i	$0.980381\pi$
$684$		0			0
$685$	691510.			0.0563083
$686$		0			0
$687$	1.10628e7			0.894283
$688$		0			0
$689$	−7.19034e6	−	1.24540e7i	−0.577034	−	0.999452i
$690$		0			0
$691$	−7.30588e6	+	1.26542e7i	−0.582073	+	1.00818i	0.413160	+	0.910658i	$0.364425\pi$
$691$	−7.30588e6	+	1.26542e7i	−0.582073	+	1.00818i	−0.995233	+	0.0975219i	$0.968908\pi$
$692$		0			0
$693$	4.28272e6	−	898040.i	0.338756	−	0.0710334i
$694$		0			0
$695$	1.68002e6	−	2.90989e6i	0.131933	−	0.228515i
$696$		0			0
$697$	−3.71464e6	−	6.43395e6i	−0.289624	−	0.501644i
$698$		0			0
$699$	566498.			0.0438536
$700$		0			0
$701$	1.90104e7			1.46115			0.730577	−	0.682830i	$-0.239251\pi$
$701$	1.90104e7			1.46115			0.730577	+	0.682830i	$0.239251\pi$
$702$		0			0
$703$	1.13066e7	+	1.95836e7i	0.862866	+	1.49453i
$704$		0			0
$705$	−204263.	+	353793.i	−0.0154780	+	0.0268088i
$706$		0			0
$707$	−7.33664e6	−	8.19328e6i	−0.552012	−	0.616466i
$708$		0			0
$709$	5.79644e6	−	1.00397e7i	0.433058	−	0.750078i	−0.564077	−	0.825722i	$-0.690768\pi$
$709$	5.79644e6	−	1.00397e7i	0.433058	−	0.750078i	0.997135	+	0.0756440i	$0.0241013\pi$
$710$		0			0
$711$	395308.	+	684693.i	0.0293266	+	0.0507951i
$712$		0			0
$713$	−1.20494e6			−0.0887653
$714$		0			0
$715$	−7.33881e6			−0.536859
$716$		0			0
$717$	−986984.	−	1.70951e6i	−0.0716989	−	0.124186i
$718$		0			0
$719$	−5.14304e6	+	8.90800e6i	−0.371020	+	0.642626i	−0.989723	−	0.142999i	$-0.954325\pi$
$719$	−5.14304e6	+	8.90800e6i	−0.371020	+	0.642626i	0.618703	+	0.785625i	$0.287659\pi$
$720$		0			0
$721$	−821420.	+	2.50466e6i	−0.0588474	+	0.179436i
$722$		0			0
$723$	1.95239e6	−	3.38163e6i	0.138906	−	0.240592i
$724$		0			0
$725$	−9.27900e6	−	1.60717e7i	−0.655626	−	1.13558i
$726$		0			0
$727$	−1.00970e7			−0.708526			−0.354263	−	0.935146i	$-0.615268\pi$
$727$	−1.00970e7			−0.708526			−0.354263	+	0.935146i	$0.615268\pi$
$728$		0			0
$729$	531441.			0.0370370
$730$		0			0
$731$	−1.35399e7	−	2.34518e7i	−0.937180	−	1.62324i
$732$		0			0
$733$	−962010.	+	1.66625e6i	−0.0661332	+	0.114546i	−0.897196	−	0.441632i	$-0.854400\pi$
$733$	−962010.	+	1.66625e6i	−0.0661332	+	0.114546i	0.831063	+	0.556178i	$0.187733\pi$
$734$		0			0
$735$	−3.05739e6	+	1.34117e6i	−0.208753	+	0.0915728i
$736$		0			0
$737$	−3.30376e6	+	5.72228e6i	−0.224047	+	0.388061i
$738$		0			0
$739$	2.10136e6	+	3.63967e6i	0.141543	+	0.245161i	0.928078	−	0.372386i	$-0.121460\pi$
$739$	2.10136e6	+	3.63967e6i	0.141543	+	0.245161i	−0.786535	+	0.617546i	$0.788127\pi$
$740$		0			0
$741$	1.66104e7			1.11131
$742$		0			0
$743$	1.99659e7			1.32684			0.663418	−	0.748249i	$-0.269105\pi$
$743$	1.99659e7			1.32684			0.663418	+	0.748249i	$0.269105\pi$
$744$		0			0
$745$	−4.00445e6	−	6.93592e6i	−0.264334	−	0.457839i
$746$		0			0
$747$	−2.85102e6	+	4.93812e6i	−0.186939	+	0.323787i
$748$		0			0
$749$	−281271.	+	857646.i	−0.0183198	+	0.0558603i
$750$		0			0
$751$	−1.81988e6	+	3.15212e6i	−0.117745	+	0.203940i	−0.918874	−	0.394552i	$-0.870900\pi$
$751$	−1.81988e6	+	3.15212e6i	−0.117745	+	0.203940i	0.801129	+	0.598492i	$0.204233\pi$
$752$		0			0
$753$	7.69992e6	+	1.33366e7i	0.494878	+	0.857155i
$754$		0			0
$755$	−4.52745e6			−0.289059
$756$		0			0
$757$	1.73429e7			1.09997			0.549986	−	0.835174i	$-0.314633\pi$
$757$	1.73429e7			1.09997			0.549986	+	0.835174i	$0.314633\pi$
$758$		0			0
$759$	−1.79156e6	−	3.10308e6i	−0.112883	−	0.195519i
$760$		0			0
$761$	5.17841e6	−	8.96927e6i	0.324142	−	0.561430i	−0.657196	−	0.753719i	$-0.728258\pi$
$761$	5.17841e6	−	8.96927e6i	0.324142	−	0.561430i	0.981338	+	0.192289i	$0.0615912\pi$
$762$		0			0
$763$	9.74238e6	+	1.08799e7i	0.605834	+	0.676572i
$764$		0			0
$765$	1.22960e6	−	2.12974e6i	0.0759647	−	0.131575i
$766$		0			0
$767$	2.96653e6	+	5.13818e6i	0.182079	+	0.315371i
$768$		0			0
$769$	−1.81548e7			−1.10707			−0.553534	−	0.832826i	$-0.686721\pi$
$769$	−1.81548e7			−1.10707			−0.553534	+	0.832826i	$0.686721\pi$
$770$		0			0
$771$	−8.85652e6			−0.536571
$772$		0			0
$773$	8.17830e6	+	1.41652e7i	0.492282	+	0.852658i	0.999960	−	0.00888861i	$-0.00282937\pi$
$773$	8.17830e6	+	1.41652e7i	0.492282	+	0.852658i	−0.507678	+	0.861547i	$0.669496\pi$
$774$		0			0
$775$	−1.66341e6	+	2.88112e6i	−0.0994823	+	0.172308i
$776$		0			0
$777$	1.11641e7	−	2.34100e6i	0.663395	−	0.139107i
$778$		0			0
$779$	−6.24627e6	+	1.08189e7i	−0.368788	+	0.638760i
$780$		0			0
$781$	−1.21124e7	−	2.09793e7i	−0.710562	−	1.23073i
$782$

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	336.q (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(4.50000 + 7.79423i) q^{3} +(-11.0358 + 19.1146i) q^{5} +(-126.882 + 26.6059i) q^{7} +(-40.5000 + 70.1481i) q^{9} +O(q^{10})\)	\(q+(4.50000 + 7.79423i) q^{3} +(-11.0358 + 19.1146i) q^{5} +(-126.882 + 26.6059i) q^{7} +(-40.5000 + 70.1481i) q^{9} +(208.355 + 360.881i) q^{11} +797.918 q^{13} -198.644 q^{15} +(687.775 + 1191.26i) q^{17} +(1156.51 - 2003.14i) q^{19} +(-778.343 - 869.224i) q^{21} +(-477.701 + 827.402i) q^{23} +(1318.92 + 2284.44i) q^{25} -729.000 q^{27} -7035.29 q^{29} +(630.596 + 1092.22i) q^{31} +(-1875.19 + 3247.93i) q^{33} +(891.689 - 2718.92i) q^{35} +(-4888.22 + 8466.65i) q^{37} +(3590.63 + 6219.16i) q^{39} -5400.95 q^{41} -19686.6 q^{43} +(-893.900 - 1548.28i) q^{45} +(1028.28 - 1781.04i) q^{47} +(15391.3 - 6751.63i) q^{49} +(-6189.98 + 10721.4i) q^{51} +(-9011.37 - 15608.2i) q^{53} -9197.45 q^{55} +20817.2 q^{57} +(3717.84 + 6439.49i) q^{59} +(-1747.69 + 3027.09i) q^{61} +(3272.39 - 9978.09i) q^{63} +(-8805.67 + 15251.9i) q^{65} +(7928.21 + 13732.1i) q^{67} -8598.62 q^{69} -58133.5 q^{71} +(-19555.3 - 33870.8i) q^{73} +(-11870.3 + 20560.0i) q^{75} +(-36038.1 - 40246.0i) q^{77} +(4880.35 - 8453.01i) q^{79} +(-3280.50 - 5681.99i) q^{81} +70395.7 q^{83} -30360.6 q^{85} +(-31658.8 - 54834.7i) q^{87} +(-72153.1 + 124973. i) q^{89} +(-101242. + 21229.3i) q^{91} +(-5675.36 + 9830.01i) q^{93} +(25526.1 + 44212.5i) q^{95} -79328.7 q^{97} -33753.5 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 36 q^{3} - 258 q^{7} - 324 q^{9}+O(q^{10}) \) 8 * q + 36 * q^3 - 258 * q^7 - 324 * q^9	\( 8 q + 36 q^{3} - 258 q^{7} - 324 q^{9} + 402 q^{11} + 924 q^{13} - 276 q^{17} + 510 q^{19} - 3564 q^{21} + 6900 q^{23} - 2814 q^{25} - 5832 q^{27} + 1080 q^{29} - 6410 q^{31} - 3618 q^{33} + 33108 q^{35} - 15250 q^{37} + 4158 q^{39} + 8616 q^{41} - 58396 q^{43} - 15060 q^{47} - 64252 q^{49} + 2484 q^{51} - 13692 q^{53} - 146248 q^{55} + 9180 q^{57} + 34830 q^{59} + 5364 q^{61} - 11178 q^{63} - 66864 q^{65} - 5994 q^{67} + 124200 q^{69} - 178536 q^{71} - 59638 q^{73} + 25326 q^{75} - 75660 q^{77} - 44062 q^{79} - 26244 q^{81} + 416892 q^{83} + 72648 q^{85} + 4860 q^{87} + 77520 q^{89} - 104722 q^{91} + 57690 q^{93} - 221376 q^{95} - 377260 q^{97} - 65124 q^{99}+O(q^{100}) \) 8 * q + 36 * q^3 - 258 * q^7 - 324 * q^9 + 402 * q^11 + 924 * q^13 - 276 * q^17 + 510 * q^19 - 3564 * q^21 + 6900 * q^23 - 2814 * q^25 - 5832 * q^27 + 1080 * q^29 - 6410 * q^31 - 3618 * q^33 + 33108 * q^35 - 15250 * q^37 + 4158 * q^39 + 8616 * q^41 - 58396 * q^43 - 15060 * q^47 - 64252 * q^49 + 2484 * q^51 - 13692 * q^53 - 146248 * q^55 + 9180 * q^57 + 34830 * q^59 + 5364 * q^61 - 11178 * q^63 - 66864 * q^65 - 5994 * q^67 + 124200 * q^69 - 178536 * q^71 - 59638 * q^73 + 25326 * q^75 - 75660 * q^77 - 44062 * q^79 - 26244 * q^81 + 416892 * q^83 + 72648 * q^85 + 4860 * q^87 + 77520 * q^89 - 104722 * q^91 + 57690 * q^93 - 221376 * q^95 - 377260 * q^97 - 65124 * q^99

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